The Hamiltonian of the system is given by H =Pkεknk, with nk = 0,1,2

Problem 1. Ultra-relativistic bosons in 2D (Points: 10+10+10 = 30)

In this problem, we consider non-interacting ultra-relativistic bosons in two dimensions (2D) in a 2D “volume” V, in contact with an external particle reservoir, and in thermal equilibrium with the surroundings. The system has a grand canonical partition function Z_g given by

Zg =^Y

k

1−e^{−β(εk−µ)−1} =e^βpV.

Here, β = 1/k_BT, wherek_B is Boltzmanns’s constant andT is temperature, while µis the chemical potential of the system. Furthermore, pis the pressure of the system. For ultra-relativistic particles, the energy of a particle in a single-particle state specified by specifying k= (kx, ky), is given by

ε_k= ¯hck.

where cis the speed of light, ¯h=h/2π, his Planck’s constant, and k ≡ |k|. The Hamiltonian of the system is given by H =^P_kε_kn_k, with n_k = 0,1,2, ....

a. Show that the pressure of the system and the average number of particles in the system,hNi, are given by

βpV = −

Z ∞

−∞dε g(ε) ln1−e^{−β(ε−µ)}, hNi = N₀+

Z ∞

−∞

dε g(ε) e^β(ε−µ)−1,

where g(ε) = ^P_kδ(ε−ε_k) and δ(x) is the δ-function. Here, N₀ is the number of particles in the lowest-energy state. Show also that the internal energy U =hHi is given by

U =

Z ∞

−∞

dε g(e) ε e^β(ε−µ)−1.

b. For this particular system, we have

g(ε) =V 2π

(hc)² ε Θ(ε), where Θ(x) = 0, x <0, Θ(x) = 1, x >0. Show that

U =K pV, and determine the purely numerical constant K.

c. Let µ → 0⁻, and compute the temperature T_λ below which we may have a macroscopic oc- cupation of the ground state (Bose-Einstein condensation).

d. In a 2D non-relativistic system of particles with mass m, with ε_k = ¯h²k²/2m, one cannot have Bose-Einstein condensation at any temperature T_λ > 0. In problem 1 c, you found a T_λ >0.

Explain on physical grounds why an ultra-relativistic non-interacting 2D boson system can feature Bose-Einstein condensation at finite temperature, while a corresponding non-relativistic system can- not.

Useful formula:

hNi= ∂lnZ_g

∂(βµ).

Problem 2. Non-interacting classical spins (Points: 10+10+10+10=40)

Consider N classical spins S_i = (S_ix, S_iy, S_iz) in a constant external magnetic field h = (0,0, h).

Let |S_i| = S be the length of the spins, which we take to be equal on all lattice sites, and ignore couplings between spins. Consider a particular case of the model, namely a system of three-state Ising spins, defined by S_i = (S_ix, S_iy, S_iz) = S(0,0, σ_i), σ_i = 0,±1.

The Hamiltonian H and the canonical partition function Z of this spin-system are given by H = −h·

N

X

i=1

Si,

Z = ^X

{Si}

e^−βH.

Here, β = 1/k_BT, wherek_B is Boltzmann’s constant and T is temperature.

a. Show that the partition functions for this spin-model is given on the form Z = (F(Sβh))^N,

where the functional form of F(x) is given by F(x) = 1 + 2 cosh(x).

b. Calculate the enthalpy,He, of the systems, where H_e =−∂lnZ

∂β .

c. Show that in general, we have for the specific heat at constant magnetic field, C_h ≡(∂H_e/∂T)_h C_h =k_Bβ^2hhH²i − hHi²ⁱ,

where hOi ≡(1/Z)^P_{Si}Oe^−βH.

d. Compute C_h in the limit Sβh1. Explain on physical grounds the result you find.

Problem 3. Classical particles in 2D (Points: 5+5+10+10=30)

Consider N non-relativistic classical particles with mass m moving in the two-dimensional (2D) (x, y)-plane with “volume” V = πR², where R is the radius of the circle to which the N particles are confined. These particles are non-interacting, but are subject to an external potential. The Hamiltonian of the system is given by

H =

N

X

i=1

"

p²_i

2m +V₀ r_i²

#

.

Here, p_i = (p_xi, p_yi) is the momentum of particle i, while r_i = ^qx²_i +y_i² is the distance from the center of the volume V of particle i. The canonical partition function Z for this system is given by

Z = 1

N!h^2N

Z

..

Z

dΓ e^−βH; dΓ =

N

Y

i=1

d²p_i d²r_i.

Here, h is a constant with dimension Js, while β = 1/k_BT, k_B is Boltzmann’s constant, and T is temperature. For later use, we also define the length R₀ ≡1/√

βV₀.

a. Show that the internal energyU =hHiin general is given by, in the canonical ensemble U =−∂lnZ

∂β .

b. Show that in this case, the canonical partition function is given

Z = 1

N! 1 λ^2N Q^N₁ ,

λ ≡ h

√2πmk_BT, Q1 ≡

Z

d²r e^−βV0r2.

c. Calculate the pressure p that the particles exert on the “walls” of the system (the perimeter of the circle) in the limits R/R₀ 1 andR/R₀ 1.

d. In polar coordinates, the Hamiltonian may be written H =

N

X

i=1

"

p²_ri

2m + p²_θi

2mr²_i +V₀r_i²

#

Here, p_ri is the radial component of the linear momentum of particlei, and p_θiis its angular momen- tum. Compute

˜

u≡ h p²_θi

2mr_i² +V0r_i²i; R/R0 1.

Useful formula:

hOi = 1 Z

1 N!h^2N

Z

..

Z

dΓ O e^−βH.

Useful formulae:

X

k

F(εk) =

Z ∞

−∞ de g(e) F(e) g(e) ≡ ^X

k

δ(e−ε_k)

Z

d^νr F(|r|) = Ω_ν

Z

dr r^ν−1 F(r); Ω_ν = 2π^ν/2 Γ(ν/2) Γ(z) ≡

Z ∞ 0

dx x^z−1 e^−x Γ(z+ 1) = z Γ(z)

ζ(z) ≡

∞

X

l=1

1 l^z

Z a 0

dx x^ν−1 e^−xν = 1 ν

Z a^ν 0

du e^−u

Z ∞ 0

dx x^z

e^x−1 = ζ(z+ 1) Γ(z+ 1)

Z ∞ 0

dx x e^−x = 1 C_h = ∂H_e

∂T

!

h

=−k_Bβ² ∂H_e

∂β

!

h

Generalized Equipartition Principle:

Let the Hamiltonian of a system be given by H =α|q|^ν +H⁰. Here q is a generalized coordinate or momentum which does not appear in H⁰. Let the partition function be given by

Z =

Z

dq

Z

dΓ⁰e^−βH, such that we have

hα|q|^νi= 1 Z

Z

dq

Z

dΓ⁰α|q|^ν e^−βH. Then we have

hα|q|^νi= k_BT ν .

Three-dimensional volume element in spherical coordinates:

d³r=dΩ r²dr; dΩ = dθ sinθ dφ Here, θ is a polar angle and φ is an azimuthal angle.